A general model of watershed extraction and representation using globally optimal ow paths and up-slope contributing areas

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1 int. j. geographical information science, 2000, vol. 14, no. 4, 337± 358 Research Article A general model of watershed extraction and representation using globally optimal ow paths and up-slope contributing areas CHAOJUN LIANG1 and D. SCOTT MACKAY2 1Environmental Remote Sensing Center, University of Wisconsin Madison, 1225 W. Dayton Street, Madison, WI, USA 2Department of Forest Ecology and Management, and Institute for Environmental Studies, University of Wisconsin Madison, 1630 Linden Dr., Room 120, Madison, WI, USA. (Received 13 July 1998; accepted 9 July 1999) Abstract. Many modern hydrological models require data inputs provided by automated digital terrain analysis functions incorporated into GIS. These inputs include elds representing surface ow directions, up-slope contributing areas, and sub-catchment partitions. Existing raster-based terrain analysis tools, including both those in oœ-the-shelf GIS packages and those in the recent literature, were designed to work with digital elevation data in mountainous topography. For highly variable topography, which may include large ood plains, lakes, wetlands, and other relatively at areas, existing tools cannot accommodate the variable signal-to-noise in the source elevation data without signi cant human intervention to handle special cases. A general model for calculating ow directions, up-slope contributing areas, and sub-catchment partitions that automatically adapts to the variable information content of grid-based elevation data sets is presented here. The model uses a combination of breadth- rst search and global optimization to extract the maximum amount of signal from any location within the data. The model is demonstrated to work well in handling topography dominated by large ood plains, lakes and other at areas without the need for a large number of empirical rules. An important contribution of the approach is the handling of explicit hydrologic features, which makes the spatial representation closely related to hydrological processes. The results have important implications for developing hydrological models that are tractable in large, heterogeneous watersheds using moderate resolution data. 1. Introduction Many distributed-components hydrological models rely heavily on GIS techniques to provide them with co-varying elds of topographic ow paths, soil hydraulic properties, and vegetation. In moderate-to-steep topography, up-slope contributing areas, surface ow directions and catchment boundaries are needed in rainfall-runoœmodelling, non-point source pollution modelling, and in distributed models of forest ecosystems. With the widespread availability of grid-based digital elevation models (DEMs) their use has become ubiquitous and GIS algorithms designed to extract hydrological properties from them almost routine. However, even the most recent research algorithms for up-slope area and ow path representation International Journal of Geographical Information Science ISSN print/issn online 2000 Taylor & Francis Ltd

2 338 C. L iang and D. Scott Mackay are limited to working in relatively small, steep catchments on clean DEMs. Although there have been methods to handle DEM noise, e.g. pits and dams, there is as yet no general method for handling all hydrologically signi cant features of large watersheds, such as wide ood plains, lakes, reservoirs, wetlands, and upland at areas. The existing body of work on this subject suggests that these features require speci c, over-parameterized models for their handling, often at the expense of diminishing the information content of the DEM ( Tarboton 1997, Mackay and Band 1998 ). We argue that such feature speci c rules hinder the development of generic, robust tools for watershed characterization. The need for a large number of parameters makes the existing tools prohibitive for modelling large watersheds at moderate resolutions. An alternative approach to terrain analysis is presented here in which global search techniques and optimization reduce the need for an excessive number of feature speci c parameters. This work is motivated by a need for improved integration of GIS and distributed hydrological modelling. Developments in GIS have lead to more e cient implementation and application of lumped and distributed simulation models. Numerous comprehensive reviews (Moore et al. 1993, Wilson 1996, Moore 1996, Paniconi et al ) describe how GIS addresses issues of data quality, data model numerical model integration, parameter aggregation, and hydrologic system representation. Our work focuses primarily on this latter issue of representation. By representation we are in part referring to the common framework within which data are collected and organized, simulation models are designed, and model results are presented. Numerous such frameworks exist. Grid-based, contour-based, TIN (triangulated irregular network) based, and purely lumped models, are example frameworks within which simulation model inputs are derived. Examples of grid-based models integrated within either commercial or custom GIS are DHSVM (Wigmosta et al ), RHESSys (Band et al ) or other models that use TOPMODEL (Beven and Kirkby 1979 ), SWAT (Arnold et al ), and Vieux et al. (1996 ). In the case of models like RHESSys and SWAT the grid-based data inputs are further organized into hydrologically signi cant features, or functional units for input to the simulations. Functional units are another part of representation that gets at the underlying conceptual representation of hydrological systems. Sub-catchments and hydrologic response units are the most common types of functional units that have been extracted from DEMs and represented within GIS. Hydrologic representation based on only catchments and hydrologic units does not adapt easily to general watershed representation in which water bodies and wide ood plains may have to be represented. By forcing the restricted representation on these areas unrealistic models may result. For example, water bodies are often represented as wide streams during watershed extraction; they are in turn simulated as wide streams in hydrologic models. Mackay and Band (1998 ) suggest that this is increasingly a common problem now that hydrologic models are applied outside of small, mountainous watersheds. Lakes are common features in the glaciated watersheds of the Upper Midwest of the United States and other areas around the Great Lakes. Reservoirs are important features in large watersheds in mountainous areas, and so they have to be represented as we scale up hydrologic modelling to a regional extent. If GIS are to support a wide variety of hydrologic modelling then it is increasingly important that automated watershed representation be adaptable to a wide range of feature types.

3 General model of watershed extraction and representation Prior results This paper addresses grid-based DEMs, which are widely available and used in a variety of applications including ow path algorithms that support hydrological modelling. The rst of these ow path algorithms were the steepest descent, or D8, algorithms of O Callaghan and Mark(1984 ) and Marks et al. (1984 ). D8 has been widely used to partition watersheds into sub-catchment areas ( Band 1986a, Jenson and Domingue 1988, Tarboton et al ), as well as calculating up-slope contributing area (Morris and Heerdegen 1988, Jenson and Domigue 1988, Band 1989, Ehlschlaeger 1989, Lammers and Band 1990, Martz and Garbrecht 1992, Moore 1992, Garbrecht and Martz 1997, Mackay and Band 1998, Wilson and Gallant 1998 ). Fractional, or F8, ow algorithms partition ow from a cell to all of its eight neighbors by weighting ow according to relative slope (Freeman 1991, Quinn et al ). Uncertainty associated with F8 weighting schemes prompted development of a ow-tube analogy in which ow across a planar surface is resolved for each cell using both aspect and gradient of the plane (Lea 1992, Costa-Cabral and Burges 1994, Tarboton 1997 ). Regardless of how ow is routed all cell-based algorithms attempt to nd surface ow directions and up-slope areas either during an ascent from concave points on the DEM or descent from convex points on the DEM. The information gathered by these algorithms is aœected by error in the DEM, such as pits or dams. Pits and dams occur as a result of insu cient or missing data during DEM production. In small, steep watersheds, pits and dams are usually negligible due to the high local topographic relief. However, in atter areas of larger watersheds or in lake-dominated areas, pit depths and dam heights often exceed local true elevation diœerences. Figure 1 shows the diœerent eœects of DEM errors in steep and at areas, respectively, and gure 2 shows how these errors propagate to ow path searching algorithms. The sensitivity of these algorithms to DEM noise results in a number of pathological drainage conditions, including gaps (Chorowicz et al. 1992, Costas-Cabral and Burges 1994 ) and loops (Band 1989, Smith et al ). A number of simple rules have been devised to overcome gaps and loops, including pit lling (Marks et al. 1984, Band 1986a, Jenson and Domingue 1988, Martz and Garbrecht 1992 ), dam breaching (Garbrecht and Martz 1996 ), and slope tolerances (Band 1989 ). During pit lling a surface is formed by lling a pit to some new pour height. This produces a surface through which ow paths can be inferred from the surrounding topography (Martz and Garbrecht 1997 ). Slope tolerances permit ow connections as long as the slope gradient of a cell is below some threshold value. They also allow for catchment area spillage over shallow divides and into adjacent catchments, and so such simple rules should be used with caution. Bennett and Armstrong (1996 ) used a bit-mapped classi cation scheme to place each DEM cell into one of six categories based on the topographic form of its local and extended neighbourhoods. Cells with similar topographic form were aggregated into networks that represented stream channels and basin divides. A similar image processing approach was rst demonstrated by Toriwaki and Fukumura (1978 ) and later extended by Band ( 1986b) for hydrological applications. Bennett and Armstrong ( 1996 ) further extended this image processing in that the stream and divide networks are extracted from the DEM and stored in vector-based data structures. To remove erroneous pits in at areas, a best rst (maximum descent/minimum ascent) method is used to nd down-slope streams. However, human intervention is needed to de ne an application speci c threshold and to decide whether to make the connection or not.

4 340 C. L iang and D. Scott Mackay Figure 1. Contrasting signal-to-noise ratios in steep and at areas of a DEM. The magnitude of errors are identical for the two areas, but dams and pits are formed in at areas. Figure 2. EŒects of errors in a DEM on the calculation of ow directions and up-slope areas. Mackay and Band (1998 ) introduced the concept of a priori identi cation of lakes and other at features on the DEM, which improved the overall catchment area and ow path calculations, and guided the selection of appropriate rules for handling these hydrologically signi cant features. Their approach allows for selective positioning of diœerent rules and tolerances within diœerent areas of the DEM, which essentially maximizes the gain of information from the data. Liang and Mackay(1997 ) re ned this approach by introducing a rule-based heuristic that minimizes the loss of DEM information content that typically occurs with the introduction of additional

5 General model of watershed extraction and representation 341 rules that require human interaction. Here we generalize and extend these results in order to develop a general tool for acquiring topographically-based hydrologic information from a wide range of watershed areas and landscape types. 3. Methods 3.1. Breadth- rst versus depth- rst search Following Marks et al. (1984 ) up-slope contributing areas are identi ed by some form of recursive upward climb from a speci ed outlet cell. From a starting cell an up-slope cell is found and the algorithms proceed by moving to the up-slope cell and continuing the search until a divide is encountered. A depth- rst search strategy is usually used by these algorithms. A depth- rst search path is rst identi ed completely to its source at a divide before any other paths are followed. Areas are accumulated on the recursive unwinding. While this is a simple and e cient way to accumulate contributing areas, it gives correct up-slope contributing areas only if the underlying ow path is identi ed correctly early in the search. However, it is not at all eœective in searching for ow paths in areas of low to moderate local relief. Unlike steep areas on a DEM ow direction information in more moderate areas is usually either disrupted by noise or unde ned (i.e., perfectly at), and so prior to accumulating the up-slope areas the ow paths must be approximated. An alternative to the depth- rst search algorithm is breadth- rst search. Figure 3 shows the search sequence of depth- rst and breadth- rst search algorithms operating in a two-dimensional grid-based search space. Depth- rst search exhausts one path before going to the next, while breadth- rst search visits all immediate neighbors of one node before proceeding to the next node. In eœect, the breadth- rst search has an advancing front line consisting of nodes held either in a queue or a stack. Initially, the front line only includes the starting nodes. After the nodes in the front line are all visited, any of their respective unvisited immediate up-slope neighbors are added to the front line and old nodes in the front line are deleted. The front line continues to expand until all nodes have been visited. Although depth- rst and breadth- rst algorithms both visit one node at a time they diœer in their search bias. Depth- rst search follows a path based on the rst node encountered in the immediate neighbourhood. Since the decision to advance is based solely on a single cell-to-cell connection the depth- rst approach is sensitive to DEM noise. However, the search pattern of breadth- rst search can be viewed as giving all nodes in the neighbourhood equal status as members of the advancing front line. The size of the queue representing the front line grows and shrinks according to the distance outward the algorithm must search in order to nd a reasonable ow path. The result of this adaptive neighborhood search is that the ow paths are not sensitive to noise as long as there is topographic information somewhere within the front line. Although the search strategies diœer they should produce identical ow directions and up-slope contributing areas in areas of high signal-to-noise, such as steep areas on DEMs. In at areas the two algorithms may result in dramatically diœerent ow paths. Flat areas are de ned as those areas on a DEM where noise disrupts some or all the relative elevations of neighbouring cells. Three distinct categories of at areas are identi ed for the purpose of our analysis. First, as a result of DEM scale, data truncation and noise, at areas in DEMs can be perfectly at, i.e. all the cells in the at area are at the same elevation. In this case, ow directions are unde ned. Second, small relative elevations of neighboring cells may be overwhelmed by DEM

6 342 C. L iang and D. Scott Mackay Figure 3. Progression through time of depth- rst search and breadth- rst search algorithms. Each square represents a node in the search tree and a cell on a grid-based DEM. Line segments represent connections between cells. The numbers in each cell represent the order in which the cells are encountered in the search. Depth- rst search tends to sweep across the search space, while breadth- rst expands outward from a starting location.

7 General model of watershed extraction and representation 343 noise. In this case, the ow directions are meaningless and are also considered unde ned. In both of the above two cases, arti cial ow paths within at areas have to be created according the elevation information of the surrounding cells for three reasons: First, in order to conform to reality as close as possible; second, in order to preserve continuity of ow paths for use in hydrological models; and last and most important, at areas occur most often within valley bottoms, and so up-slope areas would not be fully accumulated for large watersheds if at areas were not fully marked. We observe that many noisy areas on a DEM occur in the hydrologically most active areas, such as in ood plains and riparian areas. As such, it is important that ow paths through these areas be closed using the maximum amount of topographic information possible. Figure 4 shows the diœerent ow paths produced by the two search algorithms on a small piece of a at DEM. Since the cells on a DEM are connected to one another directly or indirectly, the depth rst search algorithm tends to traverse all the cells in no more than a couple of entangled paths, resulting in unrealistic ow paths. The up-slope area image accumulated with this kind of ow path will show the at areas as unrealistically wide streams where neighboring cells inside often have opposing ow directions. Since the breadth rst search algorithm depends less upon making a decision with little or no information it is not expected to produce the wide streams. The third category of at objects, which typically accounts for most of the at areas on DEMs, has low signal-to-noise with a number of cells having erroneous relative elevations, but a topographic signal exists within a larger neighbourhood. Depth- rst search often fails to nd this regional topographic signal because it cannot see more than one point-to-point connection at a time. When it changes ow direction of a potential noisy cell according to this single connection more error may be introduced. However, the advancing front line of the breadth- rst search is more tolerant of a potentially noisy cell, and so it preserves more of the original information in the DEM. The algorithm is as follows: For each cell i on the front line { For each neighbour j of i { If (gradient ( j)< at_threshold) { If ( j ows to a cell in the front line) { Do nothing; } Else { Force j to ow to i; } } } } The breadth- rst algorithm can leave a noisy cell s ow direction unchanged as long as it ows to one of the cells on the front line. Thus, only noisy cells that do not ow to the front line are considered true noise and are forced to ow to the front line. Since fewer cells are changed based on the information content of a single pair of cells, the resulting ow paths are more representative of the information content of the DEM.

8 344 C. L iang and D. Scott Mackay Figure 4. Behaviour of depth- rst search and breadth- rst search on a small at DEM. Each small square represents a DEM cell. The numbers in the cells indicate the searching sequence,and the line segments with arrow heads indicate the resulting ow directions. Depth- rst search forms long, twisted paths because cell ow directions are modi ed without considering the ow directions of surrounding cells. Breadth- rst promotes the formation of parallel ow lines because it forms preferential ow towards the starting location Globally optimal feature handling Mackay and Band ( 1998 ) represent a watershed as a cascading sequence of objects, which may be single grid cells or clumps of cells representing hydrologically distinct features. Each watershed object has methods for incorporating it into a topologically connected set of ow paths. DiŒerent object methods tune the extraction of information from the DEM. An optimization algorithm developed by Liang and Mackay ( 1997 ) performs this tuning without human intervention, but fails in at valley bottom areas and where objects straddle divides. The breadth- rst search algorithm described in the previous section overcomes the rst problem. The second problem is that objects that straddle divides must be segmented into two or more diœerent subcatchments. A solution to this is presented in this section. As with previous algorithms (Marks et al. 1984, Band 1986, 1989, Mackay and Band 1998 ), a slope threshold diœerentiates between noisy cells and clean cells for each at area. If a cell has a slope gradient that is below some threshold, it is considered a noise cell and its ow direction may be modi ed to ensure a connected set of ow paths. If the slope threshold is at or above the threshold, then the cell s drainage direction is de ned as the direction of steepest descent, in the case of D8 ow paths, or in multiple directions with ow fractionated by relative slope, in the case of F8 ow paths. In steep areas and in perfectly at areas, the slope threshold is normally set to zero. In other at areas, an optimal slope threshold must lie somewhere between zero and the maximum gradient of the area. If the threshold is set to a value below the optimum value, then the resulting ow paths may be disconnected. However, if the threshold is set too high, then more cells than necessary will be considered noisy and the drainage path will become over-connected, resulting in wide streams. We de ne the optimal threshold to be the lowest slope threshold that allows for contiguous marking of the at object. For all at areas that are not water bodies we use an optimal slope threshold. We recognize that more sophisticated methods could be developed to handle at areas, but it is our goal to have a general, adaptable approach rather than one based on handling special cases. In order to quickly nd the optimal threshold for each at object, we use a binary search method. For each at object we de ne (1) a pass threshold, T p, and (2) a fail threshold, T f. The pass threshold is initially high enough to modify the ow direction of a given cell, while the fail threshold is low enough that the cell ow

9 General model of watershed extraction and representation 345 Figure 5. DEM for the Turkey Lakes Watershed, in central Ontario. direction must be de ned using one of D8 or F8 ow. Initially, the fail threshold is set to zero and the pass threshold is set to a prede ned high value, such as the highest gradient in the area. A precision parameter, P, allows the optimization algorithm to halt when the diœerence between the pass and fail thresholds is small enough to be considered negligible. Each cell s gradient is compared against these thresholds. The gradient values are calculated as Grad(i, j)5 Ó 1 ((I(i 1, j)õ I(iÕ 1, j))21 (I(i, j1 1)Õ I(i, jõ 1))2 2N where Grad(i, j) and I(i, j) are the gradient and elevation values at row i and column j of the DEM, respectively, and N is the DEM cell size in the same unit as the elevation values. We normally set the precision to be the smallest gradient obtainable along a cardinal direction. For a DEM with one metre vertical resolution, this gives P5 1/(2N). The pass threshold T is used to mark ow paths inside a at object. If the p resulting ow paths allow all cells inside the at object to be reached, then the diœerence between T and T is checked. If the diœerence is less than P, then T is p f p the optimal threshold. Otherwise, the pass threshold may be too high, in which case it is adjusted to the half point between the pass and fail thresholds. If T does not p (1)

10 346 C. L iang and D. Scott Mackay allow for marking all cells inside the at object then it is considered too low, at which point its value is increased by 2 (T Õ T ) and the fail threshold is changed p f to the previous pass threshold. The above process is repeated until the optimal threshold is found. The following is the pseudo-code of the binary search for an optimal threshold of an object: Õ Õ P 1/(2N); T 0.05; p T 0; f while(t T > P) p f { Use breadth- rst search to mark the object with T ; p If(object all marked ) { T p } else { (T p T )/2; f tmp T ; f T T ; f p T 2(T tmp); p p } } With the above algorithm a single object s optimal threshold can be determined, but since we do not know the outlets for an object in advance, it would not be reasonable to isolate each object and optimize its threshold independently of the other objects. However, we can try to mark the whole DEM and then adjust the thresholds for all objects using the above algorithm. We then iterate until all the objects have optimal thresholds. The following is the pseudo-code of this global optimizing algorithm: bool balloptimal 5 false; while(!balloptimal) { Mark the whole DEM, each object with its individual thresholds; balloptimal 5 true; For each at object O, i { If (T of O is not optimal ) p i { Adjust T and T of O ; p f i balloptimal 5 false; } } } Watershed marking begins along the edge of the DEM. All DEM edge cells are considered possible outlets and are sorted according to their elevations, following an approach rst presented by Ehlschlaeger (1989 ). Flow paths are searched from all of the outlets starting from the lowest unmarked edge cell. Thus, objects are marked from all possible directions, which allows for their partitioning into multiple catchments when they straddle divides.

11 General model of watershed extraction and representation 347 Figure 6. DEM for the H. J. Andrews watershed, which is located in the Cascades of central Oregon. The above global optimization algorithm may produce excessively high slope thresholds for some objects that straddle divides. A divide straddling object must be marked simultaneously from all directions to be optimally marked. However, it may in fact be reached at diœerent iterations from diœerent directions if there are diœerent numbers of at objects in series within each respective catchment. Objects are optimized in the order in which they are found during an uphill climb. The divide straddling object will always be the last object found during a climb, and it will be reached rst from the direction that has the fewest objects in series. It may then be optimized from the perspective of one valley outlet while the marking processes from the other valley outlets are optimizing objects downstream. By the time the divide straddling object is reached from a second path, its threshold may already have been erroneously optimized. To solve this problem a second pass is made of all objects. In this pass each object s threshold is veri ed from all possible outlets. If the fail threshold succeeds in marking the whole object, then this object may not have been optimally marked and its optimization is repeated while holding all other objects thresholds unchanged. The second pass over the objects is relatively quick since it can use the knowledge of ow paths gained in the rst pass.

12 348 C. L iang and D. Scott Mackay Figure 7. The Up-slope area image of Ontario site produced using (a) depth- rst search without feature optimization, (b) depth- rst search with feature optimization, and (c) using breadth- rst search with global feature optimization. Water bodies, such as lakes and wetlands, are extracted using boundary marking (Mackay and Band 1998 ) rather than optimization. The boundary of a water body is marked as an absorbing boundary for all ow it receives. The area draining into the absorbing boundary is propagated to the outlet cell of the water body.

13 General model of watershed extraction and representation 349 Figure 7(b). When up-slope areas are later used to de ne a stream network and associated sub-watersheds, a topological connection is made between the outlet stream link and all incoming stream links to the water body. Topological connections are also made from surrounding subcatchment areas to the water body. This enables us to explicitly route water into water bodies, and then move water to the outlet using any amount of hydrologic sophistication as deemed necessary by speci c applications. This feature-based approach provides for a more exible representation of the watershed since it does not explicitly de ne how water moves through the water body, but it retains the water body as a topologically connected component. It has been used to clearly separate terrestrial and aquatic components of glaciated watersheds for hydrologic modelling ( Band et al ) and watershed representation (Robinson and Mackay 1996 ). 4. Results We tested our model of topographic characterization with two DEMs of contrasting topographic character. One data set is from a lake-dominated topography

14 350 C. L iang and D. Scott Mackay Figure 7(c). in the Algoma Highlands of central Ontario, Canada. The area has approximately 350 meters of relief, which is exceptional for Great Lakes area watersheds ( JeŒries et al ). Otherwise, it is characteristic of the glaciated upland areas around the Great Lakes, with a predominance of nested lakes, wetlands, and other at areas. The DEM ( gure 5) for this area was derived from a contour map using TOPOG (CSIRO 1992 ), which uses a thin plate spline interpolation algorithm (Hutchinson 1989 ). The second data set is of a 1000 km2 area within the Cascade Range in Oregon, which has over a 1000 m of relief as well as large valley bottoms and reservoirs. The data is a mosaic of USGS level 2, 7.5 minute quad DEMs ( gure 6). To test the new algorithms we used three diœerent approaches: 1. depth- rst search with no global optimization (DF) (Band, 1989 ); 2. depth- rst with global optimization (DFG) (Liang and Mackay, 1997 ); and 3. breadth- rst with global optimization (BFG). The results for the Ontario and Oregon sites are shown, respectively, in gures 7 and 8. Figure 7(a) shows the results for the Ontario DEM generated with the DF approach. In this image, entangling ow paths in the at areas form wide streams, which is a characteristic result of depth- rst search. In addition, only part of the watershed is marked. Although raising the slope threshold will increase the size of

15 General model of watershed extraction and representation 351 Figure 8. The up-slope area image of Oregon site produced using (a) depth- rst search, and (b) breadth- rst search. the marked region, it will also produce a larger number of wide streams. Figure 7(b) was generated using DFG. By isolating at areas and optimizing each non-lake at object s ow path an improved watershed representation results. The watershed is completely marked and ow pathways are clean with no erroneous wide streams. It should be noted that the DFG algorithm did not work well for all at areas. We had to change the classi cation of the two non-lake at objects near the outlet of the watershed to lake in order to avoid this problem. This result is due to the fact that the optimization criteria was designed to eliminate wide streams, which in some areas may be inconsistent with the requirements of the depth- rst search algorithm to fully mark a at area. Figure 7(c) was generated using BFG. With no modi cation to any object type, the breadth- rst search climbs through the non-lake at areas without forming entangled ow paths. One may note that some ow paths on the right edge of the main watershed extend over the divide. This is due to the quality of the DEM, which is derived from a watershed map in which areas outside of the main watershed were not as carefully mapped as areas within the watershed. As a result, the up-slope area image contains large holes. This prevents some at areas

16 352 C. L iang and D. Scott Mackay Figure 8(b). that straddle watershed ridges from being searched from all outlets, and so their slope thresholds are not fully optimized. Figure 8 shows the results for the Oregon site. Figure 8(a) was generated using the DF algorithms. Again the depth- rst search forms entangled ow paths in all at areas, and most notably in the valley bottom along the south edge of the DEM. No DFG is shown here, as it did not diœer from DF and the valley bottoms were already marked using a slope threshold of zero. Figure 8(b) was generated using BFG. No global optimization was required for this data set as there were no signi cant non-lake at objects required. In steep areas the results bear minimal diœerence to the DF approach, but in at areas the erroneous wide streams are eliminated. In the attest areas the BFG does tend to produce near-parallel ow lines, which is to be expected. In general, convergent ow lines are curvilinear and bend towards the regional ow direction. These results can be explained by examining the shape of the advancing front line as the breadth- rst search algorithm moves through the data. Figure 9 shows snapshots of the front line in the form of a greyscale image. Lighter areas in the image correspond to earlier stages in the search, while darker parts of the image correspond to later stages of search. Also shown on this gure are the positions of the front line at every 12th interval; they may be thought

17 General model of watershed extraction and representation 353 of as contours formed around pour points. Concentric search areas are formed around water bodies, which means that whole water body objects are treated as single pour points in the topography. The curvilinear ow in the large valleys can be explained by the shape of these contours, which tend to be semi-circular with their centroids positioned downstream of the contour. Once up-slope areas and ow paths are known a number of hydrologically relevant data sets can be produced e ciently. Figure 10 shows the F8 (Freeman 1991, Quinn et al ) ow paths for two selected watershed areas. F8 ow is more representative of ow in convergent topography. It is commonly used in topographybased hydrologic indices used in models based on TOPMODEL (Beven and Kirkby 1979 ). Figure 11 shows the same watershed areas partitioned into sub-catchment areas, which are used to construct databases for hydrological modelling (Band 1989, Lammers and Band 1990 ). These data products are primary inputs to many hydrologic models, including RHESSys (Band et al ) and SWAT (Arnold et al ). 5. Discussion This paper has addressed some shortcomings of existing grid-based watershed extraction and representation algorithms for low relief areas by using a combinations of object-based rule optimization and an adaptive search algorithm that is more robust than previously described algorithms. Previous image processing approaches to geomorphological feature extraction from DEMs demonstrated the potential for feature-based approaches (Toriwaki and Fukumura 1978, Band 1986b, Bennett and Armstrong 1996 ). Recent developments in feature-based terrain analysis ( Liang and Mackay 1997, Mackay and Band 1998 ) intelligently select algorithms based on feature type. However, feature-based approaches alone cannot handle highly variable topography and the need for a great many input parameters makes them ine cient in large watersheds. The approach presented here overcomes the need for excessive parameterization, by addressing the underlying problem of low signal-to-noise in relatively at areas on DEMs. The breadth- rst search algorithm presented here provides such an adapting mechanism, but does so without the need for a priori de nition of object parameters. This new model preserves as much hydrologically signi cant information as is available in the DEM. An important diœerence between depth- rst searching and breadth- rst searching needs discussion. In convergent topography, wide patches are formed using depth- rst search, but near parallel ow lines are formed by the breadth- rst search algorithm. This is consistent with the gradual reduction in signal-to-noise, which requires the algorithm to follow the trend in ow over a larger area. In wide valley bottoms this tends to produce near -parallel ow paths that are diœerentiated by a weak topographic signal. On a perfectly at surface the breadth- rst search produces parallel ow lines in the direction in which the search is initiated, as illustrated in gure 4. In contrast a depth- rst search algorithm will degenerate into a simple space- lling curve that produces a single path for the whole at surface. Recently, Garbrecht and Martz (1997 ) presented an approach to resolving ow through at areas. Their method is similar to the one presented here in that they adaptively search outward and look for drainage trends within the DEM. It diœers in that they require some criteria to determine when to stop the search outward within the at area, whereas the breadth- rst search algorithm presented here requires no such criteria. In addition, the approach here uses a single algorithm for both at

18 354 C. L iang and D. Scott Mackay Figure 9. Animation of the advancing front line of the breadth- rst search. The lighter tones indicate earlier stages of the search, while darker tones represent later stages of search. Superimposed in the image are snapshots of the position of the advancing front line at every 12 iterations. This gure shows that water bodies act as pour points around which concentric search neighbourhoods are formed. It also shows how preferential ow is formed in wide valley bottoms, such as near the lower edge of the image. The front line in these relatively at area tend to be concentric arcs, which form convergent ow lines in the downstream direction. and steep areas on the DEM, and so there is less chance of ambiguity in determining which algorithm to use as the search enters a transitional part of the landscape. 6. Conclusions With the increased application of GIS in hydrologic modelling there is a growing need for improved compatibility between the spatial representation provided by GIS and the process representation provided by models. We argued earlier that the representation of watersheds within a GIS should more closely follow processes, by explicitly recognizing hydrologic features. For example, lakes, wetlands, and large oodplains are typically viewed as problem areas for watershed extraction and representation, and yet they are salient features of most landscapes in which hydrologic studies occur. We have shown that useful information can be gained from these

19 General model of watershed extraction and representation 355 Figure 10. Fractional (F8) ow paths for two watershed areas at the Oregon site. F8 elds are a primary input to topographic indices, which are used in many hydrologic models. areas when a combination of feature-based rules and more robust search algorithms are combined. The adaptive breadth- rst searching algorithm presented here improves upon the traditional depth- rst searching algorithms in nding clean, realistic ow paths in low relief areas of DEMs. When coupled with feature-based global optimization, it derives ow path and up-slope area elds from DEMs of various source data types without the need for parameter tuning by an end-user. This demonstrates that robust and easily generalizable approaches to watershed extraction and representation can be incorporated into a GIS framework. A motivation for improving GIS-based terrain analysis tools is the need for spatial elds for input to hydrologic models of large spatial large extent, such as watersheds of order 103 km2 or larger. Tools that require considerable parameter inputs or modi cation to be adapted to diœerent parts of the landscape are inappropriate for large watershed studies at even moderate resolutions. The tools presented in this paper are appropriate for handling large watersheds with lakes, reservoirs, large ood plains, wetlands, and upland at areas. In addition, by treating these hydrologically unique parts of the landscape as features directly improves the

20 356 C. L iang and D. Scott Mackay Figure 11. Sub-catchment partitions and stream lines for two watershed areas at the Oregon site. These partitions are primary inputs to many hydrologic model, as they represent closed hydrologic systems with well-de ned boundaries. development of databases to support hydrological modelling. An important consideration for future work in this area is to better understand the sensitivity of hydrologic models to the spatial representation provided by the GIS. More eœort is needed in understanding how feature classi cation and ow path algorithm selection within GIS aœect model design and implementation. Acknowledgments Funding for this research was provided by the Graduate School of the University of Wisconsin-Madison, and McIntire-Stennis. A contour map of the Turkey Lakes Watershed, from which the DEM was generated, was generously provided by NWRI, Environment Canada. The DEM for the H. J. Andrews was provided by the Forest Science Data Bank, a partnership between the Department of Forest Science, Oregon State University, and the US Forest Service Northwest Research Station, Corvallis, Oregon. Signi cant funding for H. J. Andrews data was provided by the National Science Foundation. We thank two anonymous reviewers whose comments have improved the manuscript.

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